Multipliers and integration operators between conformally invariant spaces

In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc \D, the Besov spaces $B^p$ $(1\le p<\infty )$ and the $Q_s$ spaces $(0<s<\infty )$. Our main objective is to characterize for a given pair $(X, Y)$ of spaces in these classes, the space of pointwise multipliers $M(X, Y)$, as well as to study the related questions of obtaining characterizations of those $g$ analytic in \D such that the Volterra operator $T_g$ or the companion operator $I_g$ with symbol $g$ is a bounded operator from $X$ into $Y$.Comment: To appear in Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSA

A Hankel matrix acting on spaces of analytic functions

If $\mu$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu$ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots$, $\mu_n$ denotes the moment of order $n$ of $\mu$. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D$. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_\mu$ on Hardy spaces and on M\"obius invariant spaces.Comment: arXiv admin note: text overlap with arXiv:1612.0830

Generalized Hilbert Operators

If $g$ is an analytic function in the unit disc \D we consider the generalized Hilbert operator \hg defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in \D . More precisely, we address the question of characterizing the functions $g$ for which the operator \hg is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_\alpha$ or on the spaces of Dirichlet type $\mathcal D^p_\alpha$

Hankel matrices acting on the Hardy space H1H^1 and on Dirichlet spaces

If $\,\mu \,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_\mu \,$ be the Hankel matrix $\,(\mu _{n, k})_{n,k\ge 0}\,$ with entries $\,\mu _{n, k}=\mu _{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots$, $\mu_n\,$ denotes the moment of order $\,n\,$ of $\,\mu$. This matrix induces formally the operator $\,\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D$. When $\,\mu \,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_\mu\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1<p<\infty$, but not on $\,H^1$. J. Cima has recently proved that $\,\mathcal H\,$ is an injective bounded operator from $\,H^1\,$ into the space $\,\mathscr C\,$ of Cauchy transforms of measures on the unit circle. \par The operator $\,\mathcal H_\mu \,$ is known to be well defined on $\,H^1\,$ if and only if $\,\mu \,$ is a Carleson measure and in such a case we have that $\mathcal H_\mu (H^1)\subset \,\mathscr C$. Furthermore, it is bounded from $\,H^1\,$ into itself if and only if $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure. \par In this paper we prove that when $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure then $\,\mathcal H_\mu \,$ actually maps $\,H^1\,$ into the space of Dirichlet type $\,\mathcal D^1_0\,$. We discuss also the range of $\,\mathcal H_\mu\,$ on $\,H^1\,$ when $\,\mu \,$ is an $\alpha$-logarithmic $1$-Carleson measure ($0<\alpha <1$). We study also the action of the operators $\,\mathcal H_\mu \,$ on Bergman spaces and on Dirichlet spaces.Comment: 21 page

The event that provoked the outbreak of the so-called «Revuelta del Arrabal» according to the Muqtabis II from Ibn Hayy&#257;n Some remarks about the meaning of the passage

Una lectura atenta del texto del Muqtabis II de Ibn Hayy&#257;n que narra el acontecimiento que provocó el estallido de la llamada Revuelta del Arrabal ofrece posibilidades de interpretación que permiten dar una visión más coherente y sugestiva sobre cómo se inició este gravísimo suceso.A close reading of the text Muqtabis II from Ibn &#56256;&#56432;ayy&#257;n that deals with the event that provoked the outbreak of the so-called «Revuelta del Arrabal» («The Revolt of the Suburb» [of Córdoba]) offers new possibilities to interpret the passage in such a way that a more coherent and stimulating approach to how this event started can be given